Chapter
p
15
Mechanical Waves
PowerPoint® Lectures for
University Physics, Twelfth Edition
– Hugh
g D. Young
g and Roger
g A. Freedman
Lectures by James Pazun
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley
Goals for Chapter 15
– To study waves and their properties
– To consider wave functions and wave dynamics
– To calculate the power in a wave
– To consider wave superposition
– To
T study
t d standing
t di waves on a string
ti
Introduction
• At right, you’ll see the piles of
rubble from a highway that
absorbed just a little of the
energy from a wave
propagating through the earth
in California. In this chapter,
we’ll focus on ripples of
disturbance moving through
various media.
Waves: functions of space AND time
Definition of a Wave
Wave (from webster): "a disturbance or variation that transfers energy progressively from
point to point in a medium and that may take the form of an elastic deformation or of a
variation of pressure, electric or magnetic intensity, electric potential, or temperature."
A wave is a disturbance or variation which travels through a medium. The medium through
which the wave travels may experience some local oscillations as the wave passes, but the
particles in the medium to not travel with the wave. The disturbance may take any of a
number of shapes, from a finite width pulse to an infinitely long sine wave.
Useful website: http://www.kettering.edu/~drussell/demos.html
Types of mechanical waves
• Waves that have compressions and rarefactions parallel
to the direction of wave propagation are longitudinal.
• Waves that have compressions and rarefactions
perpendicular to the direction of propagation
propagation.
Periodic waves
• A detailed look at periodic
p
transverse waves will allow us to
extract parameters.
In Ch. 13 we oscillatory motion had only a
frequency (period) associated with it. For a wave
you have a frequency AND a wavelength (λ). The
relation
l ti b
between
t
th
them iis
f 
v

The speed of the wave, v, DOES NOT depend on f
or λ, only on the type of medium they travel on
(e.g. air, water, ground, etc.)
Periodic waves II
• A detailed look at
periodic longitudinal
waves will allow us to
extract parameters just
as we did with transverse
waves.
Mathematical description of a wave
• Wh
When th
the d
description
i ti off
the wave needs to be
more complete,
p
we can
generate a wave
function with y(x,t).
y(x  0,t)  Acost 

x 
y(x,t)
( t)  Acos
A  (t  )

v 
y(x t)  Acost  kx)
y(x,t)
2
Wave number
k


v
v
f 
or
k


x 
y(x,t)
y(
, )  Acos2 ( fft  )

 
For this example
2

 1,  1
Graphing wave functions
• Informative graphic
presentation of a wave function
is often either
y-displacement versus
x-position or
y-displacement versus x-time.
Focus on the red circles (a particular phase)

x 
y(x,t)  Acos2 ( ft  )

 
phase velocity, vphase= /T
Another velocity - material velocity*
When a wave or disturbance propagates
propagates, the particles of
the MEDIUM do not propagate, but they move a little bit
from their equilibrium positions
material velocity, vmat=dy/dt,
Material velocity is speed of the particles in the medium. If
 has dimensions of length, this is a velocity as we
normally think of it. *It’s a very bad name because waves don’t
necessarily need a medium to propagate and  might well represent
something more abstract like an electric field. But it will do for now.
Particle velocity and acceleration in a sinusoidal wave
• From the wave function,
function we have an expression for
the kinematics of a particle at any point on the wave.
Focus on the red circle that marks a particular x
If the material velocity is
perpendicular to the
phase velocity
velocity, the wave
is “transverse”.
If the material velocity
is parallel to the phase
velocity, the wave is
“longitudinal”.

2 
y(x,t)
( t)  Acos
A t 
x 
    
2
dy
(x,t)  Asint 
x 

dt
 
material velocity,
velocity vmat=dy/dt
dy/dt
How do these functions arise?
PROVIDED /k = v, a constant, they are solutions to the differential equation:
2 y(x,t)
y(x t)
 k 2 Acost  kx 
2
x
2 y(x,t)
y(x t)
  2 Acost  kx 
2
t
2
2


v 2 2 y(x,t)  2 y(x,t)
x
t
This equation results when:
•Newton’s
Newton s law is applied to a string under tension
•Kirchoff’s law is applied to a coaxial cable
•Maxwell’s equations are applied to source-free
media … and many other cases …
Speed of waves
There is a tricky part of waves that you must understand. Even though we write all
the time that v=kω=fλ, v DOES not depend on f or λ!! It only depends on the type of
medium the wave travels in.
We can derive (slowly) that in general the speed of a wave is
v
(restoring force returning the system to equilibrium)
(inertia resisting the retun to equilibrium)
A key examples is a string
v
FT

μ=M/L=linear density
The speed of a transverse wave II
• We can take a second glance at the speed of a transverse wave on a
string. How does one get it?
To derive it one looks at a small
piece of the medium (mass ∆m=μ∆x)
and writes Newton’s 2nd law. The
right hand side is the second
derivative with respect to time and
one can show (Sec. 4 in book) that
the left hand side (sum of forces in
the y-direction) yields the second
derivative with respect to position.
The constant that one obtains are
equated to the velocity (previous
slide).
Example
In the figure shown below, what is the speed of a wave in the string?
If it is shaken at 2 Hz, how many wavelengths fit in the string?
FT  msamplesg  196 N
v=
FT


FT L
 88.5
88 5 m/s
/
M string
From this velocity one gets the wavelength
v 88.5 m/s
 
-1  44.3
f
2.00 s

80
 1.81
1 81 cycles
44.3
Wave intensity and Power
Average power of a wave on a string:
Pav 
1
F  2 A 2
2
As the wave propagates outward from its source, the power generated is
spread
p
out over the area it covers. This is the intensity
y of a wave. If it is a
point source (speaker, etc.) then the intensity of a wave goes down as 1/r2
Power at position r, away from a point source
P
I(r) 
4 r 2
Area off a sphere
A
h
Example: a tornado warning siren on top of a pole
radiates sound in all directions. If the intensity is
0 250 W/m2 at a distance of 15
0.250
15.0
0 meters
meters, At what
distance is the intensity 0.01 W/m2?
I1 r12
 2  r1 
I2 r2
I1
r2  75.0
75 0 m
I2
Effects of boundaries on wave reflection
Waves in motion from one boundary (the source) to another
boundary (the endpoint) will travel and reflect.
Wave superpostion
• When waves “collide”
collide their
amplitudes add (if one is
negative and the other positive
they can add to zero)
y(x t)  y1 (x,t)
y(x,t)
(x t)  y 2 (x,t)
(x t)
Adding waves
The disturbance amplitude add and it results in a total disturbance for each
position and time.
y(x,t)  e
kx  t 
2
e
kx 10k  t 
2
Adding two waves that propagate in opposite directions and
have the same amplitude: standing waves
moves to the right
moves to the left
y(x t)  Acoskx  t  1  Acoskx  t   2 
y(x,t)
Standing waves - functions of x (only) multiplied by functions of t (only)
1
 1

y(x t)  2Acos (kx  t  1  kx  t   2 )cos (kx  t  1  kx  t   2 )
y(x,t)
2
 2





(   )
(   2 )
y(x,t)  2Acoskx  1 2 cost  1


2  
2 
Standing waves are superpositions of traveling waves
Standing waves on a string
• Fi
Fixed
d att b
both
th ends,
d th
the resonator
t was h
have waveforms
f
th
thatt match.
t h IIn
this case, the standing waveform must have nodes at both ends.
Differences arise only from increased energy in the waveform.
http://www.youtube.com/watch?v=s9GBf8y0lY0&feature=related
Normal modes (harmonics) on a string
The length of the string determines the allowed wavelengths:
2L

L  n n  n 
where n=1,2,3,….
2
n
Since the velocityy is fixed by
y the tension and the densityy of the string
g the
frequency as also discrete allowed values. These are called the harmonics
of the string (normal modes)
v
v
fn 
 fn  n
n
2L
where n=1
n=1,2,3,…
23
and
v=
FT
